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BE-CSE AND BE-IT ALGEBRA AND NUMBER THEORY SYLLABUS

August 20, 2021 0 Comments

 

ANNA UNIVERSITY, CHENNAI
AFFILIATED INSTITUTIONS

MA8551

ALGEBRA AND NUMBER THEORY

B.E-CSE AND B.E -IT

REGULATION-2017 

III-YEAR ,V-SEMESTER   

COURSE OBJECTIVE:

   *
To introduce the basic notions of groups, rings, fields which will then be used to solve
   * To introduce and apply the concepts of rings, finite fields and polynomials.
   * To understand the basic concepts in number theory
   * To examine the key questions in the Theory of Numbers.

   * To give an integrated approach to number theory an related problems.  

COURSE OUTCOME
Upon successful completion of the course, students should be able to

 * Apply the basic notions of groups, rings, fields which will then be used to solve related problems
 * Explain the fundamental concepts of advanced algebra and their role in modern mathematics and applied contexts. 
 * Demonstrate accurate and efficient use of advanced algebraic techniques.
  *  Demonstrate their mastery by solving non - trivial problems related to the concepts, and by proving simple theorems about the, statements proven by the text.
  * Apply integrated approach to number theory and abstract algebra, and provide a firm basis for further reading and study in the subject.
 
UNIT-I     GROUPS AND RINGS                                                             (12)
Groups : Definition - Properties - Homomorphism - Isomorphism - Cyclic groups - Co sets - Lagrange's theorem. Rings: Definition - Sub rings - Integral domain - Field - Integer modulo n - Ring Homomorphism
 
UNIT -II  FINITE FIELDS AND POLYNOMIALS                                    (12)
Rings - Polynomial rings - Irreducible polynomials over finite fields - Factorization of polynomials over finite fields.
 
UNIT-III   DIVISIBILITY THEORY AND CANONICAL DECOMPOSITION(12)
 Division algorithm – Base - b representations – Number patterns – Prime and composite numbers – GCD – Euclidean algorithm – Fundamental theorem of arithmetic-LCM 


UNIT-IV  DIOPHANTINE EQUATIONS AND CONGRUENCE'S         (12)
Linear Diophantine equations – Congruence‘s – Linear Congruence‘s - Applications: Divisibility tests - Modular exponentiation-Chinese remainder theorem – 2 x 2 linear systems 


UNIT V   CLASSICAL THEOREMS AND MULTIPLICATIVE FUNCTIONS (12)
Wilson‘s theorem – Fermat‘s little theorem – Euler‘s theorem – Euler‘s Phi functions – Tau and Sigma functions.
 
 
TEXTBOOKS:
 
   * Grimaldi, R.P and Ramana, B.V., "Discrete and Combinatorial Mathematics", Pearson  Education, $5^th$ Edition, New Delhi, 2007
   *  Koshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 2002. 
 
 
REFERENCES :
 
     * Lidl, R. and Pitz, G, "Applied Abstract Algebra", Springer Verlag, New Delhi, $2^nd$ Edition, 2006.
    * Niven, I., Zuckerman.H.S., and Montgomery, H.L., ―An Introduction to Theory of Numbers‖,  John Wiley and Sons , Singapore, 2004.
    * San Ling and Chaoping Xing, ―Coding Theory – A first Course‖, Cambridge Publications,     Cambridge, 2004.